15 MARCH 2001 1043 ME ´ LICE ET AL.  2001 American Meteorological Society Amplitude and Frequency Modulations of the Earth’s Obliquity for the Last Million Years J. L. ME ´ LICE* Institut de Recherche pour le De ´veloppement, Institut d’Astronomie et de Ge ´ophysique G. Lemaı ˆtre, Universite ´ Catholique de Louvain, Louvain-la-Neuve, Belgium A. CORON Institut de Physique The ´orique, Universite ´ Catholique de Louvain, Louvain-la-Neuve, Belgium A. BERGER Institut d’Astronomie et de Ge ´ophysique G. Lemaı ˆtre, Universite ´ Catholique de Louvain, Louvain-la-Neuve, Belgium (Manuscript received 19 October 1999, in final form 16 May 2000) ABSTRACT The variability of the earth’s obliquity is a result of the deviation of the obliquity from its main component at 41 kyr and is generated by amplitude and frequency modulation. For the last million years, the amplitude modulation of the obliquity explains 8% of the obliquity’s total variance and the frequency modulation explains 10.3%. The spectra of both the amplitude and frequency modulations display significant power at 171 and 97 kyr. The contribution of these two modulations to the variance of the insolation is evaluated. At 65°N in June, the amplitude modulation of the obliquity explains 1.3% of the variance of the insolation and the frequency modulation explains 1.6%. It is shown that the obliquity’s frequency modulation is physically questionable and could only be meaningful for its components at 41, 54, and 29 kyr taken separately. 1. Introduction Geological records show that for the last million years, the earth has experienced at least 10 major gla- ciations. The astronomical theory of paleoclimates (Mil- ankovitch 1941) proposes that these glaciations are the consequences of the insolation variations accompanying the perturbations of the earth’s orbit (eccentricity) and of its axis of rotation (precession of the equinoxes and obliquity). The theory finds support from the fact that the spectra of the geological records contain the same frequencies as those found in the astronomical variations with cycles at 100 (eccentricity), 41 (obliquity), and 23 and 19 (precession) kyr (Hays et al. 1976; Berger 1977; Imbrie et al. 1992). Although the eccentricity does vary with a 100-kyr period, however, the effect of this pa- *Current affiliation: Institut de Recherche pour le De ´veloppement, Laboratoire d’Oce ´anographie Dynamique et de Climatologie, Uni- versite ´ Pierre et Marie Curie, Paris, France. Corresponding author address: Prof. A. Berger, Institut d’Astronomie et de Ge ´ophysique G. Lemaı ˆtre, 2 chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium. E-mail: berger@astr.ucl.ac.be rameter on the incoming solar radiation is rather weak relative to the signals from obliquity and precession variations (Berger et al. 1993). If the response of the climate system to the insolation forcing is linear, the 100-kyr signal in the climate records should therefore be of negligible intensity, yet it is observed instead to dominate the late Pleistocene records (Imbrie et al. 1993). Nonlinear processes within the climate system and resulting from the astronomical forcing have been proposed as the origin of the 100-kyr cycle, in particular through a negative combination tone of the two main precessional periods (1/100  1/19 - 1/23) (e.g., Wig- ley 1976; Ghil and Childress 1987). Other mechanisms of internal origin (e.g., Saltzman and Verbitsky 1994) or other external forcings related to the eccentricity (Rial 1995), to the inclination of the ecliptic on a plane of reference (Muller and MacDonald 1997; Ridgwell et al. 1999) and to the obliquity have also been put forward. Within this last group, it has been proposed recently (Liu 1992, 1995, 1998; Liu and Chao 1998) that vari- ations in the frequency of the obliquity cycle could give rise to a strong 100-kyr forcing in the climate. This hypothesis is based on the fact that the spectrum of the frequency modulation of the obliquity displays periodic components close to 100 kyr.